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im looking for help on finding the local minimum and maximum of a function as well as the point of inflection

A minimum or maximum will be either at the ends of the domain. I mean where x is smalles and biggest OR

where the gradient of the tangent to the curve is 0. The gradient of the tangent is given by the first derivative. so y'=0 will give you the x value of the turning points. You have to plug that value back into the original equation to find y.

Points or inflection are where the concavit changes. This is where y''=0

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